Question: Express basis vectors $e_1 = (1, 0)$ and $e_2 = (0, 1)$ as linear combinations of $a_1= (2, - 1)$ and $a_2 = (1, 3)$.
I want to know how to solve this.
My answer is following but I am not sure about it...
$e_1= x \cdot a_1 + y \cdot a_2$ and solve for x and y
$e_2= x \cdot a_1 + y \cdot a_2$ and solve for x and y
My goal is to find the basis vectors and write the other vectors as a linear combination of the basis vectors.
Attached is the original matrix and the reduced matrix
I want to know how to do this.
Your approach works. Here's $e_1$; I'll leave $e_2$ for you.
$e_1=xa_1+ya_2$
$(1,0)=x(2,-1)+y(1,3)$
$1=2x+y$ and $0=3y-x$
$x=3y$ and $7y=1$
$y=\frac17$ and $x=\frac37$.
Check that $(1,0)=\frac37(2,-1)+\frac17(1,3)$.